Insetting a non-closed mesh into a solid object
This is going to be a bit of a tricky problem due to the edge of the crater being as rough and jagged as it is. I'm assuming that the edge of the crater is not going to line up perfectly with the tapered cylinder style you are going for.

I would go about it this way:


Create the base shape you are going for by taking a cylinder, extruding the top face a bit then scaling it down until it is an angle you like. It doesn't matter how high you extrude the face, what matters is that when you get the angle you are looking for the top face is smaller than the top edge loop of the crater. Make sure this base object is positioned where you want it to be in the final object.
Subdivide the base a lot, using CtrlR to add loops until the faces of the base are mostly square, then W and subdivide. You will want the geometry to be dense enough that the average edge on the angled portion is roughly the same as the average edge on the edge of the crater. Note that you only need to do this for the angled top faces, not the sides or bottom face of the cylinder, although it won't hurt if you do.
Look from the top at the two objects (they will need to be separate objects at least until this point) and use the circle select tool to select all of the vertices on the top of the base that the crater will cover. It doesn't have to be perfect, but you don't want there to be vertices inside the crater perimeter.
Join the two objects with CtrlJ. In edit mode, use proportional editing set to connected to adjust the height of the top edge of the base until it is fairly close to the edge of the crater. You will likely not be able to have the base be a smooth slope up to the edge of the crater due to the uneven edge. I would suggest trying to make the top of the base look a bit uneven and rocky as it gets closer to the edge, but the choice is ultimately yours.
In edit mode select the perimeter of the crater and the now uneven edge of the base using the non manifold edge method you did above.
Press CtrlE and choose 'Bridge edge loops'. The number of vertices on each edge should be close enough that it will do a decent job filling the gap. You may have to manually adjust a few places if there are any problematic spots.


Do note that I haven't tried this method on anything like your project, I'm going by what I think would work best. If you run into issues it could just be a limitation with this method, but I would imagine that it should work.